decomposition of self-adjoint elements in positive and negative parts

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decomposition

Every real valued functionf admits a well-known decomposition into its and parts: f=f+-f-. There is an analogous result for self-adjoint elements in a C*-algebra (http://planetmath.org/CAlgebra) that we will now describe.

Theorem - Let 𝒜 be a C*-algebra and a∈𝒜 a self-adjoint element. Then there are unique positive elementsa+ and a- in 𝒜 such that:

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a=a+-a-

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a+⁢a-=a-⁢a+=0

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Both a+ and a- belong to C*-subalgebra generated by a.

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∥a∥=max⁡{∥a+∥,∥a-∥}

Remark - As a particular case, the result provides a decomposition of each self-adjoint operatorT on a Hilbert space as a difference of two positive operators T=T+-T- such that Ran⁢T-⊆Ker⁢T+ and Ran⁢T+⊆Ker⁢T-, where Ran and Ker denote, respectively, the range and kernel of an operator.

Since a is , σ⁢(a)⊆ℝ, so the above functions are well defined. It is clear that

f=f+-f-⁢and⁢f+⁢f-=f-⁢f+=0⁢and⁢f+,f-⁢are both positive

(1)

The continuous functional calculus gives an isomorphismC*⁢[a]≅C0⁢(σ⁢(a)∖{0}) such that the elementa corresponds to the function f. Let a+ and a- be the elements corresponding to f+ and f- respectively. From the made in (1) it is now clear that

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a+ and a- are both positive elements.

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a=a+-a-

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a+⁢a-=a-⁢a+=0

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Both a+ and a- belong to C*⁢[a].

From the fact the every C*-isomorphism is isometric (see this entry (http://planetmath.org/HomomorphismsOfCAlgebrasAreContinuous)) and ∥f∥=max⁡{∥f+∥,∥f-∥} it follows that ∥a∥=max⁡{∥a+∥,∥a-∥}.

The uniqueness of the decomposition follows from the uniqueness of the decomposition of real valued functions in its positive and negative parts f=f+-f- (with f+⁢f-=0). □